Annals of Mathematics, 169 (2009), 741{794 The singular set of 1-1 integral currents By Tristan Riviere` and Gang Tian Abstract We prove that 2 dimensional integer multiplicity 2 dimensional rectifiable currents which are almost complex cycles in an almost complex manifold ad- mitting locally a compatible positive symplectic form are smooth surfaces aside from isolated points and therefore are J-holomorphic curves. I. Introduction 2p Let (M ;J) be an almost complex manifold. Let k 2 N, k ≤ p. We shall adopt classical notation from Geometric Measure Theory [Fe]. We say that a 2k-current C in (M 2p;J) is an almost complex integral cycle whenever it fulfills the following three conditions i) Rectifiability: There exists an at most countable union of disjoint oriented 1 1 C 2k-submanifolds C = [iNi and an integer multiplicity θ 2 Lloc(C) such that for any smooth compactly supported in M 2k-form one has X Z C( ) = θ : i Ni ii) Closedness: C is a cycle, @C = 0; i.e., 8α 2 D2k−1(M);C(dα) = 0 : iii) Almost complex: For H2k and almost every point x in C, the approximate tangent plane Tx to the rectifiable set C is invariant under the almost complex structure J; i.e., J(Tx) = Tx : In this work we address the question of the regularity of such a cycle: Does there exist a smooth almost complex manifold (Σ2k; j) without boundary and a smooth j − J-holomorphic map u (8x 2 Σ and 8X 2 TxΣ duxj · X = J · duxX) such that u would realize an embedding in M 2p aside from a locally finite 2k−2 742 TRISTAN RIVIERE` AND GANG TIAN 2k 1 2k measure closed subset of M and such that C = u∗[Σ ]; i.e., 8 2 C0 (^ M) Z C( ) = u∗ ? Σ In the very particular case where the almost complex structure J is integrable, this regularity result is optimal (C is the integral over multiples of algebraic subvarieties of M) and was established in [HS] and [Ale]. There are numer- ous motivations for studying the general case of arbitrary, almost complex structures J. First, as explained in [RT], the above regularity question for rec- tifiable almost complex cycles is directly connected to the regularity question of J-holomorphic maps into complex projective spaces. It is conjectured, for instance, that the singular set of W 1;2(M 2p;N) J-holomorphic maps between almost complex manifolds M and N should be of finite (2p − 4)-Hausdorff measure. The resolution of that question leads, for instance, to the character- ization of stable-bundle, almost complex structures over almost K¨ahlermani- folds via Hermite-Einstein Structures and extends Donaldson, Uhlenbeck-Yau characterization in the integrable case (see [Do], [UY]) to the nonintegrable one. Another motivation for studying the regularity of almost complex recti- fiable cycles is the following. In [Li] and [Ti] it is explained how the loss of compactness of solutions to geometric PDEs having a given conformal invari- ant dimension q (a dimension at which the PDE is invariant under conformal transformations - q = 2 for harmonic maps, q = 4 for Yang-Mills Fields...etc) arises along m − q rectifiable cycles (if m denotes the dimension of the do- main). These cycles happen sometimes to be almost complex (see more details in [Ri1]). 2p By trying to produce in (R ;J) an almost complex graph of real dimen- 2p sion 2k in a neighborhood of a point x0 2 R as a perturbation of a complex one (Jx0 -holomorphic), one realizes easily that, for generic almost complex structures J, the problem is overdetermined whenever k > 1 and well posed for k = 1. Therefore the case of 2-dimensional integer rectifiable almost com- plex cycles is the generic one from the existence point of view. We shall restrict to that important case in the present paper. After complexification of the tan- gent bundle to M 2p a classical result asserts that a 2-plane is invariant under J if and only if it has a 1 − 1 tangent 2-vector. Therefore we shall also speak about 1 − 1 integral cycles for the almost complex 2-dimensional integral cy- cles. In the present work we consider the locally symplectic case: We say that (M 2p;J) has the locally symplectic property if at a neighborhood of each point 2p x0 in M there exist a positive symplectic structure compatible with J and a neighborhood U of x0 and a smooth closed 2-form ! such that !(·;J·) defines a scalar product. It was proved in [RT] that arbitrary, 4-dimensional, almost complex manifolds satisfy the locally symplectic property. This is no more the case in larger dimension: one can find an almost complex structure in S6 which admits no compatible positive symplectic form even locally; see [Br]. THE SINGULAR SET OF 1-1 INTEGRAL CURRENTS 743 Our main result is the following. Theorem I.1. Let (M 2p;J) be an almost complex manifold satisfying the locally symplectic property above. Let C be an integral 2 dimensional almost complex cycle. Then, there exist a J-holomorphic curve Σ in M, smooth aside from isolated points, and a smooth integer-valued function θ on Σ such that, 1 for any 2 form 2 C0 (M), Z C( ) = θ : Σ In the \locally symplectic case" being an almost-complex 2 cycle is equiv- alent for a 2-cycle to being calibrated by the local symplectic form ! for the local metric !(·;J·). Therefore the regularity question for almost complex cy- cles is embedded into the problem of calibrated current and hence the theory of area-minimizing rectifiable 2-cycles. Therefore our result appears to be a consequence of the \Big Regularity Paper" of F. Almgren [Alm] combined with the PhD thesis of his student S. Chang [Ch]. Our attempt here is to present an alternative proof independent of Almgren's monumental work and adapted to the case we are interested in. The motivation is to give a proof that could be modified in order to solve the general case (non locally symplectic one) which cannot be \embedded" in the theory of area-minimizing cycles anymore. A proof for the regularity of almost complex cycle in the locally symplectic, p = 2 case, independent of the regularity theory for area-minimizing surfaces, was also one of the results of the work Gr=)SW of C. Taubes [Ta] for p = 2. In particular, [Ta] presents a proof of Theorem I.1 when p = 2. In [RT], we gave an alternative proof for this special case. Theorem I.1 can be seen as the generalization to higher dimension (p > 2) of these works. One of the main difficulties arising in dimension p > 2 is the nonneces- sary existence of J-holomorphic foliations transverse to our almost complex current C in a neighborhood of a point. This then prevents describing the 2 p−1 k current as a Q-multivalued graph from D into C , f(ai (z))k=1p−1 gi=1···Q in a neighborhood of a point of density N solving locally an equation of the form p−1 k X k l k (I.1) @zai = A(z; ai)l · rai + α (ai; z) ; l=1 where A and α are small in C2 norm, as for p = 2 in [RT]. What we can only ensure instead is to describe the current C, in a neighborhood of a point 2 p−1 of multiplicity Q, as an \algebraic Q-valued graph" from D into C ; that p−1 is, a family of points in C , fa1(z); : : : ; aP (z); b1(z); : : : ; bN (z)g where only P − N = Q is independent on z (neither P nor N is a priori independent on z), ai are the positive intersection points and bj are the negative ones. This \algebraic Q-valued graph" solves locally a much less attractive equation 744 TRISTAN RIVIERE` AND GANG TIAN than (I.1), p−1 p−1 k X l l X l l k (I.2) @zai = Ak(z; ai; rai) · rai + Bk(z; ai) · rai + C (z; ai) ; l=1 l=1 where A(z; a; p), B(z; a) and C(z; a) are also small in C2 norm but the depen- dence on p in A(z; a; p) is linear and therefore as rai gets bigger, which can happen, the right-hand side of (I.2) cannot be handled as a perturbation of the left-hand one in steps such as the \unique continuation argument". This was used in [RT] for proving that singularities of multiplicity Q cannot have an accumulation point in the carrier C of C. The strategy of the proof goes as follows. A classical blow-up analysis tells 2p us that, for an arbitrary point x0 of the manifold M , the limiting density −2 θ(x0) = limr!0r M(C Br(x0)). Here M denotes the mass of a current and is the restriction operator which equals π times an integer Q. Since −2 the density function r ! r M(C Br(x0)) at every point is a monotonic increasing function, the complement of the set CQ := fx 2 M ; θ(x) ≤ Qg is closed in M and this permits us to perform an inductive proof of Theorem I.1 restricting the current to CQ and considering increasing integers Q. A point of multiplicity Q is called a singular point of C if it is in the closure of points of nonzero multiplicity strictly less than Q. The goal of the proof is then to show that singularities of multiplicity less than Q are isolated.